Reservoir computing with logistic map

TL;DR

This work addresses the need for efficient reservoir computing without continuous dynamical reservoirs by using a logistic map to create virtual nodes that encode inputs into a high-dimensional state. The authors demonstrate universal applicability to both non-temporal (polynomial) and temporal (chaotic) prediction tasks, including multi-output and closed-loop scenarios, and verify robustness to moderate noise. Key contributions include accurate prediction of Lorenz, Rössler, and Hindmarsh-Rose systems, simultaneous y and z forecasting, higher-dimensional system validation, and the extension of input encoding with a simple trig-series to widen the usable parameter window. The results suggest a flexible, hardware-friendly RC approach with broad potential impact for nonlinear time-series prediction and system identification, especially when traditional continuous reservoirs are impractical.

Abstract

Recent studies on reservoir computing essentially involve a high dimensional dynamical system as the reservoir, which transforms and stores the input as a higher dimensional state, for temporal and nontemporal data processing. We demonstrate here a method to predict temporal and nontemporal tasks by constructing virtual nodes as constituting a reservoir in reservoir computing using a nonlinear map, namely the logistic map, and a simple finite trigonometric series. We predict three nonlinear systems, namely Lorenz, Rossler, and Hindmarsh-Rose, for temporal tasks and a seventh order polynomial for nontemporal tasks with great accuracy. Also, the prediction is made in the presence of noise and found to closely agree with the target. Remarkably, the logistic map performs well and predicts close to the actual or target values. The low values of the root mean square error confirm the accuracy of this method in terms of efficiency. Our approach removes the necessity of continuous dynamical systems for constructing the reservoir in reservoir computing. Moreover, the accurate prediction for the three different nonlinear systems suggests that this method can be considered a general one and can be applied to predict many systems. Finally, we show that the method also accurately anticipates the time series of the all the three variable of Rossler system for the future (self prediction).

Figures (16)

• Figure 1: The schematic diagram for training with the logistic map. The input $u_i$ is linearly transformed into $a_i$ and then is supplied into the logistic map to form virtual nodes for the reservoir. $V_i$, $i=1,2,..,L$ is the output generated from the reservoir. The iterated values of the logistic map ($\omega_k^i,~k=1,2,...,P$) are indicated as red colour dots. The $W_k,~k=1,2,...,P$, are the components of the weight matrix $W$. The black colour dotted lines inside the reservoir indicate the internal computations of the reservoir.
• Figure 2: Bifurcation diagram of the logistic map. The vertical dashed line separates the periodic region and chaotic region ($a > 3.57$). The periodic regions include period-1 region ($1.0 < a < 3.0$), period-2 cycle region ($3.0 < a < 3.449$), period-4 cycle region ($3.449< a< 3.544112$) and so on (see for example, ref.lak).
• Figure 3: Prediction of the polynomial $f(x)$ = $(x-3)(x-2)(x-1)x(x+1)(x+2)(x+3)$ for $x$ between -3 and +3 for (a) $a_{min}$ = 2.1, $a_{max}$ = 2.2, $P$ = 100, $\delta$ = 0, (b) $a_{min}$ = 2.1, $a_{max}$ = 2.2, $P$ = 100, $\delta$ = 0.1 and (c) $a_{min}$ = 3.8, $a_{max}$ = 3.9, $P$ = 20, $\delta$ = 0.1. Here $L$ = 100, $u_{min}$ = -3 and $u_{max}$ = 3.
• Figure 4: Variation of RMSE with respect to the strength of noise for the nonchaotic (red) and chaotic (blue). Here, $u_{min}$ = -3.2 and $u_{max}$ = 3.2.
• Figure 5: Prediction of Lorenz system for $y(t)$ from the input $x(t)$ for the strengths of (a) $\delta$ = 0 and (b) $\delta$ = 0.01. Here, $a_{min}$ = 1, $a_{max}$= 2, $u_{min}$ = -17, $u_{max}$ = 17, $P$ = 3 and $m$ = 100. In each of the figures, we have also indicated the Lyapunov time $t_L = t\cdot\lambda_{max}$, where $\lambda_{max}$ = 0.906 is the maximal Lyapunov exponent of the Lorenz system sprott.